Question

Tell whether the quadratic expression can be factored with integer coefficients. If it can, find the factors. $$z^{2}-26 z-87$$

Short answer

Yes, the given quadratic expression \(z^{2} - 26z - 87\) can indeed be factored with integer coefficients. The factored form of the expression is \((z+13)(z-7)\).

Step-by-step solution

Understand the Quadratic Form

Firstly, recognize that the given expression is in the quadratic form \(ax^{2} + bx + c\), where \(a = 1\), \(b = -26\) and \(c = -87\). In this case, the variable is \(z\).

Determine Factorability

In order to check if the given quadratic expression is factorable with integer coefficients, we need to check if there exists two integers \(m\) and \(n\) such that the sum of \(m\) and \(n\) equals to \(b = -26\) and the product of \(m\) and \(n\) equals to \(c = -87\). After trying out different combinations, we find that \(m = -13\) and \(n = 7\) satisfy these conditions, \( m+n = -13 + 7 =-26\) and \( m*n = -13 * 7 = -91\). Thus, this quadratic expression can be factored with integer coefficients.

Factor the Expression

Having confirmed that the expression can be factored, the next step is to express the quadratic expression in its factored form which will be \((z - m)(z - n) = (z -(-13))(z - 7) = (z+13)(z-7)\).